Mathematics

# Solve $\displaystyle\int\dfrac {\sin x}{(1+\cos x)^{2}}dx$

##### SOLUTION
$\displaystyle\int\dfrac {\sin x}{(1+\cos x)^{2}}dx$
Put, $1 + \cos x = t$

$\implies - \sin x dx = dt$

Now, $\displaystyle\int \dfrac{\sin x}{(1 + \cos x)^{2}} dx = \int \dfrac{-1}{t^2} dt$

$= -\int t^{-2} dt = - (\dfrac{t^{-1}}{-2+1}) + c$

$= \dfrac{1}{t} + c$

$= \dfrac{1}{1 + \cos x} + c$

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Subjective Medium Published on 17th 09, 2020
Questions 203525
Subjects 9
Chapters 126
Enrolled Students 84

#### Realted Questions

Q1 Single Correct Hard
If $\alpha > 1$, then $\int \dfrac {dx}{x^{2} + 2\alpha x + 1} =$.
• A. $\dfrac {1}{\sqrt {1 - \alpha^{2}}}\cot^{-1} \left (\dfrac {x + \alpha}{\sqrt {1 - \alpha^{2}}}\right ) + c$
• B. $\dfrac {1}{2\sqrt {\alpha^{2} - 1}}\log \left (\dfrac {x + \alpha - \sqrt {\alpha^{2} - 1}}{x + \alpha + \sqrt {\alpha^{2} - 1}}\right ) + c$
• C. $\dfrac {1}{2\sqrt {\alpha^{2} - 1}}\log \left (\dfrac {x + \alpha + \sqrt {\alpha^{2} - 1}}{x + \alpha - \sqrt {\alpha^{2} - 1}}\right ) + c$
• D. $\dfrac {1}{\sqrt {1 - \alpha^{2}}}\tan^{-1} \left (\dfrac {x + \alpha}{\sqrt {1 - \alpha^{2}}}\right ) + c$

1 Verified Answer | Published on 17th 09, 2020

Q2 Subjective Medium
Integrate $\dfrac { { e }^{ x }.\log { \left( \sin { { e }^{ x } } \right) } }{ \tan { \left( { e }^{ x } \right) } } dx$

1 Verified Answer | Published on 17th 09, 2020

Q3 Single Correct Medium
$\displaystyle \int 32x^{3}(\log x)^{2}dx$ is equal to
• A. $8x^{4}(\log x)^{2}+c$
• B. $x^{4}\{8(\log x)^{2}-4\log x\}+c$
• C. $x^{3}\{(\log x)^{2}+2\log x\}+c$
• D. $x^{4}\{8(\log x)^{2}-4\log x+1\}+c$

1 Verified Answer | Published on 17th 09, 2020

Q4 Single Correct Medium
$\displaystyle\int { \cfrac { 1 }{ 7 } \sin { \left( \cfrac { x }{ 7 } +10 \right) } dx }$ is equal to
• A. $\cfrac { 1 }{ 7 } \cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$
• B. $-\cfrac { 1 }{ 7 } \cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$
• C. $-7\cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$
• D. $\cos { \left( x+70 \right) } +C$
• E. $-\cos { \left( \cfrac { x }{ 7 } +10 \right) } +C$

Consider the integrals $I_1=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\frac{dx}{1+\sqrt{tan x}}$ and $I_2=\int_{\frac{\pi}{6}}^{\frac{\pi}{3}} \frac{\sqrt{sin x}dx}{\sqrt{sin }x+\sqrt{cos}x}$